Respuesta :
Answer:
0.2420 = 24.20% probability that the proportion of defective bottles in a sample of 454 bottles would be less than 12%.
Step-by-step explanation:
I am going to use the binomial approximation to the normal to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]n = 454, p = 0.13[/tex]
So
[tex]\mu = E(X) = np = 454*0.13 = 59[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{454*0.13*0.87} = 7.1657[/tex]
What is the probability that the proportion of defective bottles in a sample of 454 bottles would be less than 12%?
12% of 454 is 0.12*454 = 54.48.
The number of defective bottles is a discrete value, so you can only have 54 or 55 defective bottles. 54 is less than 12%, so this probability is the pvalue of Z when X = 54.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{54 - 59}{7.1657}[/tex]
[tex]Z = 0.7[/tex]
[tex]Z = 0.7[/tex] has a pvalue of 0.7580
1 - 0.7580 = 0.2420
0.2420 = 24.20% probability that the proportion of defective bottles in a sample of 454 bottles would be less than 12%.
